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Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

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Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem mathematics Article Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem Ruslan Gabdullin 1, Vladimir Makarenko 1 and Irina Shevtsova 1,2,3,4,* 1 Department of Mathematical Statistics, Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991 Moscow, Russia; [email protected] (R.G.); [email protected] (V.M.) 2 Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou 310018, China 3 Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences, 119333 Moscow, Russia 4 Moscow Center for Fundamental and Applied Mathematics, 119991 Moscow, Russia * Correspondence: [email protected] Abstract: Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones. Keywords: central limit theorem; Lindeberg’s theorem; normal approximation; asymptotically exact Citation: Gabdullin, R.; Makarenko, constant; asymptotically best constant; uniform distance; Lindeberg fraction; truncated moment; V.; Shevtsova, I. Asymptotically Exact absolute constant Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem. Mathematics 2021, 9, 501. https://doi.org/10.3390/math9050501 1. Introduction In various applications of probability theory, one has to approximate an unknown Academic Editor: Antonio Di distribution of a sum of independent random variables with some known law. Such prob- Crescenzo lems arise, for example, in insurance, financial mathematics, reliability theory, queueing theory, and many other areas. The most common approximation is the normal one which Received: 6 February 2021 is based on the central limit theorem. The adequacy of the normal approximation can be Accepted: 24 February 2021 Published: 1 March 2021 estimated with the help of convergence rate estimates in the central limit theorem such as the celebrated Berry–Esseen [1,2] inequality (in terms of full moments and under the addi- Publisher’s Note: MDPI stays neutral tional moment-type assumptions), or Osipov–Petrov’s [3,4], Esseen’s [5], Rozovskii’s [6], with regard to jurisdictional claims in Wang-Ahmad’s [7] inequalities and their generalizations [8–11] (in terms of truncated published maps and institutional affil- moments without any additional assumptions). However, the most natural estimates, such iations. as Esseen’s, Rozovskii’s and Wang–Ahmad’s inequalities contained unknown constants, and their application in practice was made possible only by the results of [8,11], where in particular, the unknown constants in the above inequalities were evaluated. A detailed overview of the cited inequalities can be found in [11] and for brevity, we do not duplicate it here. Since the crucial role in estimation of the adequacy of the normal approximation Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. is played by the values (upper bounds, in fact) of appearing absolute constants, it is very This article is an open access article important to understand how accurate the existing upper bounds for the constants are, distributed under the terms and how much they might be lowered and if it is worth trying to improve the method of their conditions of the Creative Commons evaluation. The problem becomes much deeper as soon as we observe that estimates of the Attribution (CC BY) license (https:// accuracy of the normal approximation are usually used with large sample sizes or when creativecommons.org/licenses/by/ the majorizing expressions are assumed to be small, so that, in fact, not only the absolute 4.0/). values of the appearing constants are of interest, but also their presumably more optimistic Mathematics 2021, 9, 501. https://doi.org/10.3390/math9050501 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 501 2 of 32 (smaller) values which may be used under the corresponding asymptotic assumptions. Every set of asymptotic assumptions generates the corresponding asymptotic constant. Hence, we can introduce a whole classification of asymptotic constants. The present study is devoted to investigation of the various asymptotic constants and the main purpose is to construct their lower bounds. Let X1, X2, ... , Xn be independent random variables (r.v.’) with distribution functions 2 (d.f.’s) Fk(x) = P(Xk < x), x 2 R, expectations EXk = 0, variances sk = VarXk, k = 1, ... , n, and such that n 2 2 Bn := ∑ sk > 0. k=1 For n = 1, 2, . denote n Sn − ESn Xk Sn = X1 + X2 + ··· + Xn, Sen = p = ∑ , VarSn k=1 Bn Z x 1 −t2/2 F(x) = p e dt, x 2 R, Dn = Dn(F1,..., Fn) = sup P(Sen < x) − F(x) , 2p −¥ x2R 2 2 3 sk (z) = EXk 1(jXkj > z), mk(z) = EXk 1(jXkj < z), k = 1, . , n 1 n 1 n ( ) = ( ) = E 3 (j j < ) Mn z : 3 ∑ mk zBn 3 ∑ Xk 1 Xk zBn , Bn k=1 Bn k=1 1 n ( ) = Ej j3 (j j < ) Ln z : 3 ∑ Xk 1 Xk zBn , Bn k=1 1 n 1 n ( ) = 2( ) = E 2 (j j ) > Ln z 2 ∑ sk zBn 2 ∑ Xk 1 Xk > zBn , z 0. Bn k=1 Bn k=1 The function Ln( · ) is called the Lindeberg fraction. It is easy to see that jMn(z)j 6 Ln(z), z > 0. In case of independent identically distributed (i.i.d.) r.v.’s X1, ... , Xn we denote their common d.f. by F and write Dn(F) := Dn(F,..., F). In [8] it was proved that Dn 6 AE(#, g) sup fgjMn(z)j + zLn(z)g, (1) 0<z<# Dn 6 AR(#, g) gjMn(#)j + sup zLn(z) , #, g > 0, n 2 N, (2) 0<z<# where the functions AE(#, g), AR(#, g) depend only on # and g (that is, they turn into absolute constants as soon as # and g are fixed), both are monotonically non-increasing with respect to g > 0, and AE(#, g) is also non-increasing with respect to # > 0. The question on the boundedness of AR(#, g) as # ! ¥ is still open, while AE(0+, g) = AR(0+, g) = ¥ for every g > 0 and AE(#, 0+) = AR(#, 0+) = ¥ for every # > 0. To avoid ambiguity, in what follows by constants appearing here in various inequalities we mean their exact values; in particular, in majorizing expressions — their least possible values. Upper bounds for the constants AE(#, g) and AR(#, g) for some # and g computed in [8] are presented in Tables1 and2, respectively. Here the symbol g∗ stands for the point of minimum of the upper bound for AR(#, g), obtained within the framework of the method used in [8], i.e., the upper bound for AR(#, g) found in [8] remains constant as g > g∗ grows for every fixed # > 0. More precisely, the quantity g∗ is defined as follows: p p 2 4 1/4 g∗ = 1/ 6{ = (1 − t + t ) /(t 3) = 0.5599 . , Mathematics 2021, 9, 501 3 of 32 where t 2 (p/2, p) is the unique root of the equation tan t = t/(1 − t2), q −2 2 2 2 { = x (cos x − 1 + x /2) + (sin x − x) = 0.5315 . , x=x0 x0 = 5.487414 . is the unique root of the equation 8(cos x − 1) + 8x sin x − 4x2 cos x − x3 sin x = 0, x 2 (p, 2p). Table 1. Two-sided bounds for the constants AE(#, g) from inequality (1), g∗ = 0.5599 ... Upper bounds were obtained in [8], and the lower ones in Theorem3 below. # g AE(#, g) 6 AE(#, g) > # g AE(#, g) 6 AE(#, g) > 1.21 0.2 2.8904 0.5006 ¥ 1 2.6588 0.3703 1.24 0.2 2.8900 0.4889 ¥ 0.97 2.6599 0.3736 ¥ 0.2 2.8846 0.4876 2.56 ¥ 2.6500 0 1.76 0.4 2.7360 0.4606 2.62 5 2.6500 0.2126 5.94 0.4 2.7300 0.4606 2.65 4 2.6500 0.2117 ¥ 0.4 2.7299 0.4606 2.74 3 2.6500 0.2458 1 g∗ 2.7367 0.5795 3.13 2 2.6500 0.3018 1.87 g∗ 2.6999 0.4359 4 1.62 2.6500 0.3222 ¥ g∗ 2.6919 0.4359 5.37 1.5 2.6500 0.3294 1 0.72 2.7298 0.5746 ¥ 1.43 2.6500 0.3339 1 ¥ 2.7286 0 ¥ ¥ 2.6409 0 4.35 1 2.6600 0.3703 0+ 8 ¥ ¥ Table 2. Two-sided bounds for the constants AR(#, g) from inequality (2), g∗ = 0.5599 ... Upper bounds were obtained in [8], and the lower ones in Theorem3 below. # g AR(#, g) 6 AR(#, g) > # g AR(#, g) 6 AR(#, g) > 1.21 0.2 2.8700 0.5048 1.99 g∗ 2.6600 0.4357 5.39 0.2 2.8635 0.5000 2.12 g∗ 2.6593 0.4397 1.76 0.4 2.6999 0.4485 3 g∗ 2.6769 0.4586 2.63 0.4 2.6933 0.4675 5 g∗ 2.7562 0.4784 0.5 g∗ 3.0396 1.1329 0+ 8 ¥ ¥ 1 g∗ 2.7286 0.5795 In the same paper [8] there were found sharpened upper bounds for the constants AE(#, g) and AR(#, g) provided that the corresponding fractions LE,n(#, g) := sup fgjMn(z)j + zLn(z)g, LR,n(#, g) := gjMn(#)j + sup zLn(z).
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